# Find the value of k for which the simultaneous equation x + y + z = 3; x + 2y + 3z = 4 and x + 4y + kz = 6 will not have a unique solution.

A.0

B.5

C.6

D.7

Answer

Verified

381.6k+ views

Hint: We need to have a basic idea of solving the system of equations in three variables to solve this problem. Use the determinant of a matrix to solve this problem.

The given equations are

x + y + z = 3

x + 2y + 3z = 4

x + 4y + kz = 6

we can represent the given system of equations in matrix form using the coefficients of the variables.

$ \Rightarrow \left| {\matrix

1 & 1 & 1 \\

1 & 2 & 3 \\

1 & 4 & k \\

\endmatrix } \right|$

The given system of equations will be consistent with unique solution, when

$\left| {\matrix

1 & 1 & 1 \\

1 & 2 & 3 \\

1 & 4 & k \\

\endmatrix } \right| \ne 0$

Finding the determinant of the above matrix, we get

$ \Rightarrow 1(2k - 12) + 1(3 - k) + 1(4 - 2) \ne 0$

On simplification,

$ \Rightarrow k - 12 + 3 + 2 \ne 0$

$ \Rightarrow k - 7 \ne 0$

$ \Rightarrow k \ne 7$

For k = 7, the given simultaneous equations will not have a unique solution. Hence option D is the correct answer.

Note:

To solve a system of equations we have different methods available: substitution method, graph method, elimination method. The system of equations in three variables are either dependent, independent or inconsistent. Dependent systems of equations have an infinite number of solutions. Independent system of equations has only one solution. Inconsistent systems of equations have no solution. If the determinant of a matrix is zero it represents a linearly dependent system.

The given equations are

x + y + z = 3

x + 2y + 3z = 4

x + 4y + kz = 6

we can represent the given system of equations in matrix form using the coefficients of the variables.

$ \Rightarrow \left| {\matrix

1 & 1 & 1 \\

1 & 2 & 3 \\

1 & 4 & k \\

\endmatrix } \right|$

The given system of equations will be consistent with unique solution, when

$\left| {\matrix

1 & 1 & 1 \\

1 & 2 & 3 \\

1 & 4 & k \\

\endmatrix } \right| \ne 0$

Finding the determinant of the above matrix, we get

$ \Rightarrow 1(2k - 12) + 1(3 - k) + 1(4 - 2) \ne 0$

On simplification,

$ \Rightarrow k - 12 + 3 + 2 \ne 0$

$ \Rightarrow k - 7 \ne 0$

$ \Rightarrow k \ne 7$

For k = 7, the given simultaneous equations will not have a unique solution. Hence option D is the correct answer.

Note:

To solve a system of equations we have different methods available: substitution method, graph method, elimination method. The system of equations in three variables are either dependent, independent or inconsistent. Dependent systems of equations have an infinite number of solutions. Independent system of equations has only one solution. Inconsistent systems of equations have no solution. If the determinant of a matrix is zero it represents a linearly dependent system.

Recently Updated Pages

Which of the following would not be a valid reason class 11 biology CBSE

What is meant by monosporic development of female class 11 biology CBSE

Draw labelled diagram of the following i Gram seed class 11 biology CBSE

Explain with the suitable examples the different types class 11 biology CBSE

How is pinnately compound leaf different from palmately class 11 biology CBSE

Match the following Column I Column I A Chlamydomonas class 11 biology CBSE

Trending doubts

The lightest gas is A nitrogen B helium C oxygen D class 11 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Which place is known as the tea garden of India class 8 social science CBSE

What is pollution? How many types of pollution? Define it

Write a letter to the principal requesting him to grant class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Draw a diagram of a plant cell and label at least eight class 11 biology CBSE